Galois module in nLab

• Idea
• Properties
• Examples
• Related concepts
• References

Proposition

Let K↪LK\hookrightarrow L be a Galois extension of a number field KK.

Then the ring of integers O LO_L of this extension is a Galois module of Gal(K↪L)Gal(K\hookrightarrow L).

(see also Hilbert-Speiser theorem?)

Example

(ll-adic representation)

Let ll be a prime number. Let Gal(k↪k¯)Gal(k\hookrightarrow \overline k) be the absolute Galois group of a number field kk. Then a morphism of groups

Gal(k↪k¯)→Aut(M)Gal(k\hookrightarrow \overline k)\to Aut (M)

is called an ll-adic representation of Gal(k↪k¯)Gal(k\hookrightarrow \overline k). Here MM is either a unit dimensional vector space over the algebraic closure ℚ¯ l\overline \mathbb{Q}_l or a finitely generated module over the integral closure ℤ¯ l\overline \mathbb{Z}_l.

In particular the ll-adic Tate-module is of this kind.

Example

(ll-adic Tate module) Let ll be a prime number. Let AA be an abelian group. The ll-adic Tate module is defined to be the limit

T l(A)=lim nker(l n)T_l(A)=lim_n \;ker (l^n)

i.e. it is the limit over the directed diagram ker(p n+1)→ker(p n)ker(p^{n+1})\to ker(p^n). Here the kernel ker(p n)ker(p^n) of the multiplication-with-p np^n map p n:A→Ap^n:A\to A is called p np^n-torsion of AA.

Example

(the Tate-module)

Let k Sk_S denote the separable closure of kk. Let AA be the group of roots of unity of k sk_s in kk. Then the ll-adic Tate-module of the absolute Galois group Gal(k↪k s)Gal(k\hookrightarrow k_s) is called the ll-adic Tate module of kk or the ll-adic cyclotomic character of kk.

It is equivalently the Tate-module of the multiplicative group scheme μ k\mu_k.

The Tate-module is endowed with the structure of a ℤ\mathbb{Z}-module by z(a n) n=((zmodulop n)a n) nz(a_n)_n=((z\; modulo\; p^n)a_n)_n.

Example

(ll-adic Tate module of an abelian variety)

Let ll be a prime number. Let GG be an abelian variety over a field kk. Let k sk_s denote the separable closure of kk. The k sk_s-valued points of GG assemble to an abelian group.

Then there are classical results on the rank of the Tate-module T l(G)T_l(G): For example if the characteristic of kk is a prime number p≠lp\neq l we have that T l(G)T_l(G) is a free ℤ l\mathbb{Z}_l module of rank 2dim(G)2dim(G).

A special case of the Tate conjecture can be formulated via Tate-modules:

Let kk be finitely generated over its prime field of characteristic p≠lp\neq l. Let A,BA,B be two abelian varieties over kk. Then the conjecture states that

hom(A,B)⊗ℤ p≃hom(T l(A),T l(B))hom(A,B)\otimes \mathbb{Z}_p\simeq hom(T_l(A),T_l(B))

If kk is a finite field or a number field the conjecture is true.

Example

(l-adic cohomology of a smooth variety)

Let ll be a prime number. Let XX be a smooth variety? over a field kk of characteristic prime to ll. Let k sk_s denote the separable closure of kk.

The ll-adic cohomology in degree ii is defined to be the directed limit lim nH et i(X k s,ℤ/l nℤ)lim_n\; H^i_{et}(X_{k_s}, \mathbb{Z}/l^n\mathbb{Z}). It is a Galois module where the action is given by pullback.

More specifically, given σ∈Gal(k↪k s)\sigma\in Gal(k\hookrightarrow k_s) it acts on the X k s=X⊗ kk sX_{k_s}=X\otimes_k k_s via the second factor. This is an isomorphism, since σ\sigma is an automorphism, and hence σ *\sigma^* on cohomology is an isomorphism.

Note that since we have a equivalence T lA≃H et 1(A k s,ℤ l) ∨T_l A\simeq H_{et}^1(A_{k_s}, \mathbb{Z}_l)^\vee, we have that the l-adic Tate module is a special case of the ll-adic cohomology.